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Existence and nonexistence of hypercyclic semigroups


Authors: L. Bernal-González and K.-G. Grosse-Erdmann
Journal: Proc. Amer. Math. Soc. 135 (2007), 755-766
MSC (2000): Primary 47A16; Secondary 47D03
DOI: https://doi.org/10.1090/S0002-9939-06-08524-8
Published electronically: August 31, 2006
MathSciNet review: 2262871
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Abstract: In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from--and considerably shorter than--the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Fréchet space. This complements recent results due to Bès and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.


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  • 1. S. I. Ansari, Existence of hypercyclic operators on topological vectors spaces, J. Funct. Anal. 148 (1997), 384-390. MR 1469346 (98h:47028a)
  • 2. T. Bermúdez, A. Bonilla, J. A. Conejero and A. Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math. 170 (2005), 57-75. MR 2142183 (2005m:47013)
  • 3. T. Bermúdez, A. Bonilla and A. Martinón, On the existence of chaotic and hypercyclic semigroups on Banach spaces, Proc. Amer. Math. Soc. 131 (2003), 2435-2441. MR 1974641 (2004b:47006)
  • 4. T. Bermúdez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria, Bull. Austral. Math. Soc. 70 (2004), 45-54. MR 2079359 (2005d:47014)
  • 5. L. Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), 1003-1010. MR 1476119 (99f:47010)
  • 6. L. Bernal-González and A. Bonilla, Exponential type of hypercyclic entire functions, Arch. Math. (Basel) 78 (2002), 283-290. MR 1895500 (2002m:30033)
  • 7. L. Bernal-González and K.-G. Grosse-Erdmann, The Hypercyclicity Criterion for sequences of operators, Studia Math. 157 (2003), 17-32. MR 1980114 (2003m:47013)
  • 8. J. Bès and K. Chan, Denseness of hypercyclic operators on a Fréchet space, Houston J. Math. 29 (2003), 195-206. MR 1952504 (2003j:47006)
  • 9. J. Bès and K. Chan, Approximation by chaotic operators and by conjugate classes, J. Math. Anal. Appl. 284 (2003), 206-212. MR 1996128 (2004j:47014)
  • 10. J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), 94-112. MR 1710637 (2000f:47012)
  • 11. J. Bonet, L. Frerick, A. Peris and J. Wengenroth, Transitive and hypercyclic operators on locally convex spaces, Bull. London Math. Soc. 37 (2005), 254-264. MR 2119025 (2005k:47021)
  • 12. J. Bonet, F. Martínez-Giménez and A. Peris, A Banach space which admits no chaotic operator, Bull. London Math. Soc. 33 (2001), 196-198. MR 1815423 (2001m:47015)
  • 13. J. Bonet, F. Martínez-Giménez and A. Peris, Linear chaos on Fréchet spaces, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 1649-1655. MR 2015614 (2004i:47016)
  • 14. J. Bonet and A. Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), 587-595. MR 1658096 (99k:47044)
  • 15. K. C. Chan and J. H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991), 1421-1449. MR 1142722 (92m:47060)
  • 16. J. A. Conejero Casares, ``Operadores y semigrupos de operadores en espacios de Fréchet y espacios localmente convexos", Ph.D. thesis, Universitat Politècnica de València, Valencia, 2004.
  • 17. J. A. Conejero and A. Peris, Linear transitivity criteria, Topology Appl. 153 (2005), 767-773. MR 2203889
  • 18. G. Costakis and M. Sambarino, Topologically mixing hypercyclic operators, Proc. Amer. Math. Soc. 132 (2004), 385-389. MR 2022360 (2004i:47017)
  • 19. W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems 17 (1997), 793-819. MR 1468101 (98j:47083)
  • 20. L. Dunford and J. T. Schwartz, ``Linear Operators", Part I, Interscience, New York, 1958. MR 0117523 (22:8302)
  • 21. S. El Mourchid, On a hypercyclicity criterion for strongly continuous semigroups, Discrete Contin. Dyn. Syst. 13 (2005), 271-275. MR 2152389 (2006d:47017)
  • 22. K.-J. Engel and R. Nagel, ``One-parameter semigroups for linear evolution equations", Springer, New York, 2000. MR 1721989 (2000i:47075)
  • 23. R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288. MR 0884467 (88g:47060)
  • 24. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. MR 1111569 (92d:47029)
  • 25. S. Grivaux, Construction of operators with prescribed behaviour, Arch. Math. (Basel) 81 (2003), 291-299. MR 2013260 (2004g:47011)
  • 26. S. Grivaux, Hypercyclic operators, mixing operators, and the bounded steps problem, J. Operator Theory 54 (2005), 147-168. MR 2168865
  • 27. K.-G. Grosse-Erdmann, On the universal functions of G. R. MacLane, Complex Variables Theory Appl. 15 (1990), 193-196. MR 1074061 (91i:30021)
  • 28. K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), 345-381. MR 1685272 (2000c:47001)
  • 29. K.-G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math. 139 (2000), 47-68. MR 1763044 (2001f:47051)
  • 30. K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), 273-286. MR 2068180 (2005c:47010)
  • 31. D. W. Hadwin, E. A. Nordgren, H. Radjavi and P. Rosenthal, Most similarity orbits are strongly dense, Proc. Amer. Math. Soc. 76 (1979), 250-252. MR 0537082 (80i:47029)
  • 32. J. Horváth, ``Topological vector spaces and distributions", vol. I, Addison-Wesley, Reading, 1966. MR 0205028 (34:4863)
  • 33. C. Kitai, ``Invariant closed sets for linear operators'', Ph.D. thesis, University of Toronto, Toronto, 1982.
  • 34. F. León-Saavedra, Notes about the hypercyclicity criterion, Math. Slovaca 53 (2003), 313-319. MR 2025025 (2004k:47010)
  • 35. F. León-Saavedra and A. Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), 524-545. MR 1469352 (98h:47028b)
  • 36. F. Martínez-Giménez and A. Peris, Chaos for backward shift operators, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 12 (2002), 1703-1715. MR 1927407 (2003h:47056)
  • 37. V. Müller, ``Spectral theory of linear operators and spectral systems in Banach algebras", Birkhäuser Verlag, Basel, 2003. MR 1975356 (2004c:47008)
  • 38. G. A. Muñoz, Y. Sarantopoulos and A. Tonge, Complexifications of real Banach spaces, polynomials and multilinear maps, Studia Math. 134 (1999), 1-33. MR 1688213 (2000g:46009)
  • 39. R. I. Ovsepian and A. Pe\lczynski, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $ L^2$, Studia Math. 54 (1975), 149-159. MR 0394137 (52:14942)
  • 40. J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874-920. MR 0005803 (3:211b)
  • 41. A. Peris, Hypercyclicity criteria and the Mittag-Leffler theorem, Bull. Soc. Roy. Sci. Liège 70 (2001), 365-371. MR 1904062 (2003c:47012)
  • 42. A. Peris and L. Saldivia, Syndetically hypercyclic operators, Integral Equations Operator Theory 51 (2005), 275-281. MR 2120081 (2005h:47017)
  • 43. S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. MR 0241956 (39:3292)
  • 44. W. Rudin, ``Functional Analysis'', 2nd ed., McGraw-Hill, New York, 1991. MR 1157815 (92k:46001)

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Additional Information

L. Bernal-González
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Apdo. 1160, Avda. Reina Mercedes, 41080 Sevilla, Spain
Email: lbernal@us.es

K.-G. Grosse-Erdmann
Affiliation: Fachbereich Mathematik, Fernuniversität Hagen, 58084 Hagen, Germany
Email: kg.grosse-erdmann@fernuni-hagen.de

DOI: https://doi.org/10.1090/S0002-9939-06-08524-8
Keywords: Hypercyclic uniformly continuous semigroup of operators, topologically mixing semigroup, Hypercyclicity Criterion, supercyclic semigroup
Received by editor(s): November 29, 2004
Received by editor(s) in revised form: October 5, 2005
Published electronically: August 31, 2006
Additional Notes: The first author was partially supported by Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 and by Ministerio de Ciencia y Tecnología Grant BFM2003-03893-C02-01
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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